Symbolic Logic, Dale Jacquette, Wadsworth, 2001, Chpt. 8, IV(6), p.435

A whimiscal argument to round off a rather uninspiring week:

Something is rotten in the state of Denmark only if all 3-day-old herrings are putrifying. There are at least some 3-day-old herrings if there are at least some 3-day-old mackerels. Thus, we may conclude that something rotten in the state of Denmark is a 3-day-old mackerel only if something is putrifying.

The works:

1. (∃x)Rxd ⊃(x)[(Tx • Hx) ⊃ Px]
2. (∃x)(Tx • Mx) ⊃(∃x)(Tx • Hx)
3. ∴(∃x)(Rxd • Tx • Mx) ⊃(∃x)Px
4. * (∃x)(Rxd • Tx • Mx) ......... ACP
5. * Rad • Ta • Ma ......... 4 EI x/a
6. * Ta • Ma ......... 5 Simp.
7. * (∃x)(Tx • Mx) ......... 6 EG
8. * (∃x)(Tx • Hx) ......... 2,7 MP
9. * Tm • Hm ......... 8 EI x/m
10. * Rad ......... 5 Simp.
11. * (∃x)Rxd ......... 10 EG
12. * (x)[(Tx • Hx) ⊃ Px] ......... 11,1 MP
13. * (Tm • Hm) ⊃ Pm ......... 12 UI x/m
14. * Pm ......... 9,13 MP
15. * (∃x)Px ......... 14 EG
16. (∃x)(Rxd • Tx • Mx) ⊃(∃x)Px ......... 4-15 CP